Generally, to obtain (REF ), we applied only one Markov’s
inequality. Thus, on the left hand side of (REF ), we have the sum of \(\mathrm {F}^{\textup {Env}} (\cdot )\)  and \(\mathrm {F}^{\textup {Task}} (\cdot )\) .
A relaxed form of (REF ), can be obtained by applying the affine transformation
and also Markov’s inequality two times at the task and
environment levels, separately. As discussed in (REF ), the
affine transformation leads to a new function denoted by \(\mathrm {G} \left(\right)\) .
For example, if the convex function is \(\mathrm {F} \left(a,b\right)=(a-b)^2\) , since from \(\mathrm {F} \left(a,b\right)=(a-b)^2\le c\) , we conclude that \(a\le b+\sqrt{c}\) , the affine transformation leads to \(k=1\)  and \(\mathrm {G} \left(c\right)=\sqrt{c}\) . Similarly, \(\mathrm {F} \left(a,b\right)=kl(a,b)\le c\)  leads to \(k=1/(1-0.5\lambda )\) , and \(\mathrm {G} \left(c\right)=c/(\mathrm {M}.\lambda (1-0.5\lambda ))\)  [1]} .
The
following corollary is a relaxation of (REF ).
